On the Life and Work of S. Helgason

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egv geomet- a gave he 17)adasister a and -1973) c gru Helgason’s igurður su nAkureyri in asium raino choice. of ortation 00(oyih holder) (copyright 0000 aSigurðardóttir ra nnorthern in , on. 2 G.ÓLAFSSONANDR.J.STANTON was established in 1930 and was the second Gymnasium in Iceland. There Helgason studied mathematics, physics, languages, amongst other subjects during the years 1939-1945. He then went to the University of Iceland in Reykjavík where he enrolled in the school of engineering, at that time the only way there to study mathematics. In 1946 he began studies at the University of Copenhagen from which he received the Gold Medal in 1951 for his work on Nevanlinna-type value distribution theory for analytic almost-periodic functions. His paper on the subject became his master’s thesis in 1952. Much later a summary appeared in [H89]. Leaving Denmark in 1952 he went to Princeton University to complete his graduate studies. He received a Ph.D. in 1954 with the thesis, Banach Algebras and Almost Periodic Functions, under the supervision of Salomon Bochner. He began his professional career as a C.L.E. Moore Instructor at M.I.T. 1954- 56. After leaving Princeton his interests had started to move towards two areas that remain the main focus of his research. The first, inspired by Harish-Chandra’s ground breaking work on the representation theory of semisimple Lie groups, was Lie groups and harmonic analysis on symmetric spaces; the second was the Radon transform, the motivation having come from reading the page proofs of Fritz John’s famous 1955 book Plane Waves and Spherical Means. He returned to Princeton for 1956-57 where his interest in Lie groups and symmetric spaces led to his first work on applications of Lie theory to differential equations, [H59]. He moved to the University of Chicago for 1957-59, where he started work on his first book [H62]. He then went to Columbia University for the fruitful period 1959-60, where he shared an office with Harish-Chandra. In 1959 he joined the faculty at M.I.T. where he has remained these many years, being full professor since 1965. The periods 1964-66, 1974-75, 1983 (fall) and 1998 (spring) he spent at the Institute for Advanced Study, Princeton, and the periods 1970-71 and 1995 (fall) at the Mittag-Leffler Institute, Stockholm. He has been awarded a degree Doctoris Honoris Causa by several universities, notably the University of Iceland, the University of Copenhagen and the University of Uppsala. In 1988 the American Mathematical Society awarded him the Steele Prize for expository writing citing his book Differential Geometry and Symmetric Spaces and its sequel. Since 1991 he carries the Major Knights Cross of the Icelandic Falcon.

2. Mathematical Research In the Introduction to his selected works, [Sel], Helgason himself gave a personal description of his work and how it relates to his published articles. We recommend this for the clarity of exposition we have come to expect from him as well as the insight it provides to his motivation. An interesting interview with him also may be found in [S09]. Here we will discuss parts of this work, mostly those familiar to us. We start with his work on invariant diﬀerential operators, continuing with his work on Radon transforms, his work related to symmetric spaces and representation theory, then a sketch of his work on wave equations.

2.1. Invariant Differential Operators. Invariant differential operators have always been a central subject of investigation by Helgason. We find it very infor- mative to read his first paper on the subject [H59]. In retrospect, this shines a beacon to follow through much of his later work on this subject. Here we find a ONTHELIFEANDWORKOFS.HELGASON 3 lucid introduction to differential operators on manifolds and the geometry of homogeneous spaces, reminiscient of the style to appear in his famous book [H62]. Specializing to a reductive homogeneous space, he begins the study of D(G/H), those differential operators that commute with the action of the group of isome- tries. The investigation of this algebra of operators will occupy him through many years. What is the relationship of D(G/H) to D(G) and what is the relationship of D(G/H) to the center of the universal enveloping algebra? Harish-Chandra had just described his isomorphism of the center of the universal enveloping algebra with the Weyl invariants in the symmetric algebra of a Cartan subalgebra, so Hel- gason introduces this to give an alternative description of D(G/H). But the goal is always to understand analysis on the objects, so he investigates several problems, variations of which will weave throughout his research. For symmetric spaces X = G/K the algebra D(X) was known to be commu- tative, and Godement had formulated the notion of harmonic function in this case obtaining a mean value characterization. Harmonic functions being joint eigenfunctions of D(X) for the eigenvalue zero, one could consider eigenfunctions for other eigenvalues. Indeed, Helgason shows that the zonal spherical functions are also eigenfunctions for the mean value operator. When X is a two-point homogeneous space, and with Ásgeirsson’s result on mean value properties for solutions of the ultrahyperbolic Laplacian in Euclidean space in mind, Helgason formulates and proves an extension of it to these spaces. Here D(X) has a single generator, the Laplacian, for which he constructs geometrically a fundamental solution, thereby allowing him to study the inhomogeneous problem for the Laplacian. This paper contains still more. In many ways the two-point homogeneous spaces are ideal generalizations of Euclidean spaces so following F. John [J55] he is able to define a Radon like transform on the constant curvature ones and identify an inversion operator. Leaving the Riemannian case, Helgason considers harmonic Lorentz space G/H. He shows D(G/H) is generated by the natural second order operator; he obtains a mean value theorem for suitable solutions of the generator and an explicit inverse for the mean value operator. Finally, he examines the wave equation on harmonic Lorentz spaces and shows the failure of Huygens principle in the non-flat case. Building on these results he subsequently examines the question of existence of fundamental solutions more generally. He solves this problem for symmetric spaces as he shows that every D ∈ D(X) has a fundamental solution, [H64, Thm. 4.2]. ∞ ′ Thus, there exists a distribution T ∈ Cc (X) such that DT = δxo . Convolution ∞ then provides a method to solve the inhomogeneous problem, namely, if f ∈ Cc (X) then there exists u ∈ C∞(X) such that Du = f. Those results had been announced in [H63c]. The existence of the fundamental solution uses the deep results of Harish-Chandra on the aforementioned isomorphism as well as classic results of Hörmander on constant coefficient operators. It is an excellent example of the combination of the classical theory with the semisimple theory. Here is a sketch of his approach. In his classic paper [HC58] on zonal spherical functions, Harish-Chandra introduced several important concepts to handle harmonic analysis. One was the appropriate notion of a Schwartz-type space of K bi-invariant functions, there denoted I(G). I(G) with the appropriate topology is a Fréchet space, and having ∞ K Cc (X) as a dense convolution subalgebra. 4 G.ÓLAFSSONANDR.J.STANTON

Another notion from [HC58] is the Abel transform

ρ Ff (a)= a f(an) dn . ZN Today this is also called the ρ-twisted Radon transform and denoted Rρ. Eventually Harish-Chandra showed that this gives a topological isomorphism of I(G) onto S(A)W , the Weyl group invariants in the Schwartz space on the Euclidean space A. Furthermore, the Harish-Chandra isomorphism γ : D(X) → D(A) interacts compatibly in that Rρ(Df)= γ(D)Rρ(f) . One can restate this by saying that the Abel transform turns invariant differential equations on X into constant coefficient differential equations on A ≃ a ≃ RrankX . t ′ W ′ It follows then that Rρ : S (A) → I(G) is also an isomorphism. This can then be used to pull back the fundamental solution for γ(D) to a fundamental solution for D. The article [H64] continues the line of investigation from [H59] into the structure of D(X). If we denote by U(g) the universal enveloping algebra of gC, then U(g) is isomorphic to D(G). Let Z(G) be the center of D(G). This is the algebra of bi-invariant differential operators on G. The algebra of invariant differential operators on X is isomorphic to D(G)K /D(G)K ∩ D(G)k and therefore contains Z(G) as an Abelian subalgebra. Let h be a Cartan subalgebra in g extending a and denote by Wh its Weyl group. The subgroup Wh(a) = {w ∈ Wh | w(a) = a} induces the little Weyl group W by W W restriction. It follows that the restriction p 7→ p|a maps S(h) h into S(a) . Now C Z(G) ≃ S(g )G ≃ S(h)Wh , and D(X) ≃ S(a)W ≃ S(s)K , s a Cartan complement of k. The structure of these various incarnations is given in cf. [H64, Prop. 7.4] and [H92, Prop 3.1]. See also the announcements in [H62a, H63c]: Theorem 2.1. The following are equivalent: (1) D(X)= Z(G). C W C W (2) S(h ) h |a = S(a ) . G K (3) S(g) |s = S(s) . A detailed inspection showed that (2) was always true for the classical symmetric spaces but fails for some of the exceptional symmetric spaces. Those ideas played an important role in [ÓW11] as similar restriction questions were considered for sequences of symmetric spaces of increasing dimension. The final answer, prompted by a question from G. Shimura, is [H92]: Theorem 2.2. Assume that X is irreducible. Then Z(G)= D(X) if and only if X is not one of the following spaces E6/SO(10)T, E6/F4, E7/E6T or E8/E7SU(2). Moreover, for any irreducible X any D ∈ D(X) is a quotient of elements of Z(G). 2.2. The Radon Transform on Rn. The Radon transform as introduced by J. Radon in 1917 [R17] [RaGes] associates to a suitable function f : R2 → C its integrals over affine lines L ⊂ R2

R(f)(L)= f(L) := f(x) dx Zx∈L for which he derived an inversion formula.b This ground breaking article appeared in a not easily available journal (one can find the reprinted article in [H80]), and ONTHELIFEANDWORKOFS.HELGASON 5 consequently was not well known. Nevertheless, its true worth is easily determined by the many generalizations of it that have been made in geometric analysis and representation theory, some already pointed out in Radon’s original article. An important milestone in the development of the theory was F. John’s book [J55]. Later, the application of integration over affine lines in three dimensions played an important role in the three dimensional X-ray transform. We refer to [E03, GGG00, H80, H11, N01] for information about the history and the many applications of the Radon transform and its descendants. Helgason first displayed his interest in the Radon transform in that basic paper [H59]. There he considers a transform associated to totally geodesic submanifolds in a space of constant curvature and produces an inversion formula. To use it as a tool for analysis one needs to determine if there is injectivity on some space of rapidly decreasing functions and compatibility with invariant differential operators, just as Harish-Chandra had done for the map Ff . In[H65] Helgason starts on his long road to answering such questions, and, in the process recognizing the under- lying structure as incidence geometry, he is able to describe a vast generalization. As he had previously considered two-point homogeneous spaces he starts there, but to this he extends Radon’s case to affine p-planes in Euclidean space. We summarize the results in the important article [H65]. Denote by H(p,n) the space of p-dimensional affine subspaces of Rn. Let f ∈ ∞ Rn Cc ( ) and ξ ∈ H(p,n). Define

R(f)(ξ)= f(ξ) := f(x) dξx Zx∈ξ where the measure dξx is determinedb in the following way. The connected Euclidean motion group E(n) = SO(n) ⋉Rn acts transitively on both Rn and H(p,n). Take Rn basepoints xo = 0 ∈ and ξo = {(x1,...,xp, 0,..., 0)} ∈ H(p,n) and take dξo x Lebesgue measure on ξo. For ξ ∈ H(p,n) choose g ∈ E(n) such that ξ = g · ξo. ∗ Then dξx = g dξo x or

f(x)dξx = f(g · x) dx . Zξ Zξo For x ∈ Rn the set x∨ := {ξ ∈ H(p,n) | x ∈ ξ} is compact, in fact isomorphic to the Grassmanian G(p,n) = SO(n)/S(O(p)×O(n−p)) of all p-dimensional subspaces of Rn. Therefore each of these carries a unique SO(n)-invariant probability measure dxξ which provides the dual Radon transform. Let ϕ ∈ Cc(Ξ) and deﬁne

∨ ϕ (x)= ϕ(ξ) dxξ . ∨ Zx We have the Parseval type relationship

f(ξ)ϕ(ξ) dξ = f(x)ϕ∨(x) dx Rn ZΞ Z and both the Radon transformb and its dual are E(n) intertwining operators. If p = n − 1 every hyperplane is of the form ξ = ξ(u,p)= {x ∈ Rn | hx, ui = p} n−1 R and ξ(u,p)= ξ(v, q) if and only if (u,p)= ±(v, q). Thus H(p,n) ≃ S ×Z2 . We now we have the hyperplane Radon transform considered in [H65]. This case had been considered by F. John [J55] and he proved the following inversion formulas 6 G.ÓLAFSSONANDR.J.STANTON for suitable functions f:

n−1 1 1 2 f(x) = n−1 ∆x f(u, hu, xi) du, n odd 2 (2πi) n−1 ZS n−2 1 2 ∂bpf(u,p) f(x) = n ∆x dpdu , n even . (2πi) n−1 R p − hu, xi ZS Z The difference between the even and odd dimensionsb is significant, for in odd dimensions inversion is given by a local operator, but not in even dimension. This is fundamental in Huygens’ principle for the wave equation to be discussed subsequently. For Helgason the problem is to show the existence of suitable function spaces on which these transforms are injective and to show they are compatible with the E(n) invariant differential operators. One shows that the Radon transform extends to the Schwartz space S(Rn) of rapidly decreasing functions on Rn and it maps that space into a suitably defined Schwartz space S(Ξ) on Ξ. Denote by D(Rn), respectively D(Ξ), the algebra of E(n)-invariant differential operators on Rn, respectively Ξ . 2 Furthermore, define a differential operator on Ξ by f(u, r)= ∂r f(u, r). However a new feature arises whose existence suggests future difficulties in generalizations. Let

S∗(Rn)= {f ∈ S(Rn) | f(x)p(x) dx =0 for all polynomials p(x)} Rn Z and S∗(Ξ) = {ϕ ∈ S(Ξ) | ϕ(u, r)q(r) dr =0 for all polynomials q(r)} . R Z Finally, let SH (Ξ) be the space of rapidly decreasing function on Ξ such that for each k ∈ Z+ the integral ϕ(u, r)rk dr can be written as a homogeneous polynomial in u of degree k. Then we have the basic theorem for this transform and its dual: R Theorem 2.3. [H65] The following hold: (1) D(Rn)= C[∆] and D(Ξ) = C[]. (2) ∆f = f. n (3) The Radon transform is a bijection of S(R ) onto SH (Ξ) and the dual n transformc b is a bijection SH (Ξ) onto S(R ). (4) The Radon transform is a bijection of S∗(Rn) onto S∗(Ξ) and the dual transform is a bijection S∗(Ξ) onto S∗(Rn). (5) Let f ∈ S(Rn) and ϕ ∈ S∗(Ξ). If n is odd then f = c ∆(n−1)/2(f)∨ and ϕ = c (n−1)/2(ϕ∨)∧ for some constant independent of f and ϕ. (6) Let f ∈ S(Rn) and ϕ ∈ Sb∗(Ξ). If n is even then ∨ ∨ ∧ f = c1 J1(f) and ϕ = c2 J2(ϕ )

where the operators J1 and J2 are given by analytic continuation b α J1 : f(x) 7→ an.cont|α=1−2n f(y)kx − yk dy Rn Z and β J2 : ϕ 7→ an.cont|β=−n ϕ(u, r)ks − rk dr R Z ONTHELIFEANDWORKOFS.HELGASON 7

and c1 and c2 are constants independent of f and g. In [H80] it was shown that the map

f 7→ (n−1)/4 f extends to an isometry of L2(Rn) onto L2(Ξ) . b Needed for the proof of the theorem is one of his fundamental contributions to the subject in the following support theorem in [H65]. An important generalization of this theorem will be crucial for his later work on solvability of invariant diﬀerential operators on symmetric spaces. Theorem 2.4 (Thm 2.1 in [H65]). Let f ∈ C∞(Rn) satisfy the following conditions: (1) For each integer x 7→ kxkk|f(x)| is bounded. (2) There exists a constant A> 0 such that f(ξ)=0 for d(0, ξ) > A. Then f(x)=0 for kxk > A. b An important technique in the theory of the Radon transform, which also plays an important role in the proof of Theorem 2.3, uses the Fourier slice formula: Let r> 0 and u ∈ Sn−1 then

F(f)(ru)= c f(u,s)e−isr ds . (2.1) R Z n Rn So that if f is supported in a closed ballb Br (0) in of radius r centered at the origin, then by the classical Paley-Wiener theorem for Rn the function r 7→ F(f)(ru) extends to a holomorphic function on C such that sup(1 + |z|)ne−r|Imz||F(f)(zu) < ∞ z∈C ∞ Let Cr,H (Ξ) be the space of ϕ ∈ SH (Ξ) such that p 7→ ϕ(u,p) vanishes for p > r. Then the Classical Paley-Wiener theorem combined with (2.1) shows that the ∞ Rn ∞ Radon transform is a bijection Cr ( ) ≃ Cr,H (Ξ), [H65, Cor. 4.3]. (2.1) also played an important role in Helgason’s introduction of the Fourier transform on Riemannian symmetric spaces of the noncompact type.

2.3. The Double Fibration Transform. The Radon transform on Rn and the dual transform are examples of the double ﬁbration transform introduced in [H66b, H70]. Recall that both Rn and H(p,n) are homogeneous spaces for the group G = E(n). Let K = SO(n), L = S(O(p) × O(n − p)) and N = p n {(x1,...,xp, 0,... 0) | xj ∈ R} ≃ R and H = L ⋉ N. Then R ≃ G/K, H(p,n) ≃ G/H and L = K ∩ H. Hence we have the double ﬁbration

G/L (2.2) t ❏❏ π tt ❏❏ p tt ❏❏ tt ❏❏ ty t ❏$ X = G/K Ξ= G/H 8 G.ÓLAFSSONANDR.J.STANTON where π and p are the natural projections. If ξ = a · ξo ∈ Ξ and x = b · xo ∈ X then

f(ξ)= f(ah · xo) dH/L(hL) (2.3) ZH/L and b ∨ ϕ (x)= ϕ(bk · ξo) dK/L(kL) (2.4) ZK/L for suitable normalized invariant measures on H/L ≃ N and K/L. More generally, using Chern’s formulation of integral geometry on homogeneous spaces as incidence geometry [C42], Helgason introduced the following double fibration transform. Let G be a locally compact Hausdorff topological group and K,H two closed subgroups giving the double fibration in 2.2. We will assume that G, K, H and L := K ∩ H are all unimodular. Therefore each of the spaces X = G/K, Ξ= G/H, G/L, K/L and H/L carry an invariant measure. We set xo = eK and ξo = eH. Let x = aK ∈ X and ξ = bH ∈ Ξ. We say that x and ξ are incident if aK ∩ bH 6= ∅. For x ∈ X and ξ ∈ Ξ we set xˆ = {η ∈ Ξ | x and ξ are incident } and similarly ξ∨ = {x ∈ X | ξ and x are incident } . Assume that if a ∈ K and aH ⊂ HK then a ∈ H and similarly, if b ∈ H and bK ⊂ KH then b ∈ K. Thus we can view the points in Ξ as subsets of X, and similarly points in X are subsets of Ξ. Then x∨ is the set of all ξ such that x ∈ ξ and ξˆ is the set of points x ∈ X such that x ∈ ξ. We also have −1 ∨ −1 xˆ = p(π (x)) = aK · ξ0 ≃ H/L and ξ = π(p (ξ)) = bH · xo ≃ K/L. Under these conditions the Radon transform (2.3) and its dual (2.4) are well defined at least for compactly supported functions. Moreover, for a suitable normalization of the measures we have fˆ(ξ)ϕ(ξ) dξ = f(x)ϕ∨(x) dx . ZΞ ZX Helgason [H66b, p.39] and [GGA, p.147] proposed the following problems for these transforms f → fˆ, ϕ → ϕ∨: (1) Identify function spaces on X and Ξ related by the integral transforms f 7→ f and ϕ 7→ ϕ∨. (2) Relate the functions f and f ∨ on X, and similarly ϕ and (ϕ∨)∧ on Ξ, includingb an inversion formula, if possible. (3) Injectivity of the transformsb and description of the image. (4) Support theorems. (5) For G a Lie group, with D(X), resp. D(Ξ), the algebra of invariant differential operators on X, resp. Ξ. Do there exist maps D 7→ D and E 7→ E∨ such that (Df)∧ = Df and (Eϕ)∨ = E∨ϕ∨ . b There are several examples where the double fibration transform serves as a guide, e.g. the Funk transform onb theb sphere Sn, see [F16] for the case n =2, and more generally [R02]; and the geodesic X-ray transform on compact symmetric spaces, see [H07, R04]. Other uses of the approach can be found in [K11]. We refer the reader to [E03] and [H11] for more examples. ONTHELIFEANDWORKOFS.HELGASON 9

2.4. Fourier analysis on X = G/K. From now on G will stand for a noncompact connected semisimple Lie group with finite center and K a maximal compact subgroup. We take an Iwasawa decomposition G = KAN and use standard notation for projections on to the K and A component. Set X = G/K as before and denote by xo the base point eK. Given Helgason’s classic presentation of the structure of symmetric spaces [H62] there is no good reason for us to repeat it here, so we use it freely and we encourage those readers new to the subject to learn it there. In this section we introduce Helgason’s version of the Fourier transform on X, see [H65a, H68, H70]. At first we follow the exposition in [ÓS08] which is based more on representation theory, i.e. à la von Neumann and Harish-Chandra, rather than geometry as did Helgason. For additional information see the more modern representation theory approach of [ÓS08], although we caution the reader that in some places notation and definitions differ. 2 −1 The regular action of G on L (X) is ℓgf(y) = f(g · y), g ∈ G and y ∈ X. 1 For an irreducible unitary representation (π, Vπ) of G and f ∈ L (X) set

π(f)= f(g)π(g) dg . ZG Here we have pulled back f to a right K-invariant function on G. If π(f) 6=0 then K Vπ = {v ∈ Vπ | (∀k ∈ K) π(k)v = v} is nonzero. Furthermore, as (G, K) is a K Gelfand pair we have dim Vπ =1, in which case (π, Vπ) is called spherical. K Fix a unit vector eπ ∈ Vπ . Then Tr(π(f)) = (π(f)eπ,eπ) and kπ(f)kHS = kπ(f)eπk. Note that both (π(f)eπ,eπ) and kπ(f)eπk are independent of the choice of eπ. Let GK be the set of equivalence classes of irreducible unitary spherical representations of G. Then as G is a type one group, there exists a measure µ on GK such thatb b 2 2 bf(g · xo)= (π(f)eπ, π(g)eπ)dµ(π) and kfk = kπ(f)eπkHS dµ(π) . (2.5) b 2 b ZGK ZGK Harish-Chandra, see [HC54, HC57,b HC58, HC66], determinedb the representations that occur in the support of the measure in the decomposition (2.5), as well as an explicit formula for the Plancherel measure for the spherical Fourier transform defined by him. Helgason’s formulation is motivated by “plane waves". First we fix parameters. ∗ Let (λ, b) ∈ aC × K/M and define an “exponential function” eλ,b : X → C by

λ−ρ eλ,b(x)= eb(x) ,

−1 2 where eb(x)= a(x b) from the Iwasawa decomposition. Let Hλ = L (K/M) with action −1 πλ(g)f(b)= eλ,b(g · xo)f(g · b) .

It is easy to see that πλ is a representation with a K-ﬁxed vector eλ(b)=1 for all b ∈ K/M; there is a G-invariant pairing

C Hλ × H−λ¯ → , hf,gi := f(b)g(b) db ; (2.6) ZK/M 10 G.ÓLAFSSONANDR.J.STANTON and it is unitary if and only if λ ∈ ia∗ and irreducible for almost all λ [K75, H76].

With fλ := πλ(f)eλ we have

efλ(b)= πλ(f)eλ(b)= f(g)πλ(g)eλ(b) dg = f(x)eλ,b(x) dx . (2.7) ZG ZX Then fe(λ, b) := fiλ(b) is the Helgason Fourier transform on X, see [H65a, Thm 2.2]. Recalle the littlee Weyl group W . The representation πwλ is known to be equiv- ∗ alent with πλ for almost all λ ∈ aC. Hence for such λ there exists an intertwining operator

A(w, λ): Hλ → Hwλ . The operator is unique, up to scalar multiples, by Schur’s lemma. We normalize it so that A(w, λ)eλ = ewλ. The family {A(w, λ)} depends meromorphically on λ and A(w, λ) is unitary for λ ∈ ia∗. Our normalization implies that

A(w, λ)fλ = fwλ . (2.8) We can now formulate the Plancherel Theorem for the Fourier transform in the e e following way, see [H65a, Thm 2.2] and also [H70, p. 118]. First let c(λ)= a(¯n)−λ−ρ dn¯ ¯ ZN be the Harish-Chandra c-function, λ in a positive chamber. The Gindikin- Karpele- vich formula for the c-function [GK62] gives a meromorphic extension of c to all ∗ ∗ of aC. Moreover c is regular and of polynomial growth on ia . To simplify the notation let dµ(λ,kM) be the measure (#W |c(λ)|)−1 dλd(kM) on ia∗ × K/M.: Theorem 2.5 ([H65a]). The Fourier transform establishes a unitary isomorphism ⊕ 2 dλ L (X) ≃ (πλ, Hλ) . ∗ |c(λ)|2 Zia /W ∞ Furthermore, for f ∈ Cc (X) we have

f(x)= fλ(b)e−λ,b(x) dµ(λ, b) . ∗ Zia ×K/M Said more explicitly, the Fourier transforme extends to a unitary isomorphism W L2(X) → L2 ia∗,dµ,L2(K/M) 2 ∗ 2 = ϕ∈ L ia ,dµ,L (K/M ) (∀w ∈ W )A(w, λ)ϕ(λ)= ϕ(wλ) . To connect it with Harish-Chandra’s spherical transform notice that if f is left

K-invariant, then b 7→ fλ(b)= f(λ) is independent of b and the integral 2.7 can be written as e e f(λ)= eλ,b(x) db dx = f(x)ϕλ(x) dx (2.9) ZX ZK/M ZX where ϕλ is the sphericale function

−1 λ−ρ ϕλ(x)= a(g k) dk . ZK ONTHELIFEANDWORKOFS.HELGASON 11

Then (2.9) is exactly the Harish-Chandra spherical Fourier transform [HC58] and the proof of Theorem 2.5 can be reduced to that formulation. Since ϕλ = ϕµ if and only if that there exists w ∈ W such that wλ = µ, the spherical Fourier transform f(λ) is W invariant. The Plancherel Theorem reduces to e Theorem 2.6. The spherical Fourier transform sets up an unitary isomorphism dλ L2(X)K ≃ L2 ia∗/W, . |c(λ)|2 K If f ∈ Cc(X) then 1 dλ f(x)= f(λ)ϕ−λ(x) . #W ∗ |c(λ)|2 Zia A very related result is the Paley-Wienere theorem which describes the image of the smooth compactly supported functions by the Helgason Fourier Transform. For K-invariant functions in [H66] Helgason formulated the problem and solved it modulo an interchange of a specific integral and sum. The justification for the interchange was provided in [G71]; a new proof was given in [H70, Ch.II Thm. 2.4]. ∞ The Paley-Wiener theorem for functions in Cc (X) was announced in [H73a] and a complete proof was given in [H73b, Thm. 8.3]. Later, Torasso [T77] produced another proof, and Dadok [D79] generalized it to distributions on X. There are many applications of the Paley-Wiener Theorem and the ingredients of its proof. For example an alternative approach to the inversion formula can be obtained [R77]. The Paley-Wiener theorem was used in [H73b] in the proof of surjectivity discussed in the next section, and in [H76] to prove the necessary and sufficient condition for the bijectivity of the Poisson transform for K-finite functions on K/M to be discussed subsequently. The Paley-Wiener theorem plays an important role in the study of the wave equation on X as will be discussed later. For the group G, an analogous theorem, although much more complicated in statement and proof, was finally obtained by Arthur [A83], see also [CD84, CD90, vBS05]. In [D05] the result was extended to non K-finite functions. The equivalence of the apparently different formulations of the characterization can be found in [vBSo12]. For semisimple symmetric spaces G/H it was done by van den Ban and Schlichtkrull [vBS06]. The local Paley-Wiener theorem for compact groups was derived by Helgason’s former student F. Gonzalez in [G01] and then for all compact symmetric spaces in [BÓP05, C06, ÓS08, ÓS10, ÓS11].

2.5. Solvability for D ∈ D(X). We come to one of Helgason’s major results: a resolution of the solvability problem for D ∈ D(X). We have seen the existence of a fundamental solution allows one to solve the inhomogeneous equation: given ∞ ∞ ∞ f ∈ Cc (X) does there exists u ∈ C (X) with Du = f? But what if f ∈ C (X)? This is much more difficult. Given Helgason’s approach outlined earlier it is natural that once again he needs a Radon-type transform but more general than for K bi- invariant functions. The Radon transform on symmetric spaces of the noncompact type is, as men- tioned in the earlier section, an example of the double fibration transform and probably one of the motivating examples for S. Helgason to introduce this general framework. Here the double fibration is given by 12 G.ÓLAFSSONANDR.J.STANTON

G/M (2.10) ss ▲▲▲ π ss ▲▲p sss ▲▲▲ sy ss ▲▲& X = G/K Ξ= G/MN and the corresponding transforms are for compactly supported functions:

∨ f(g · ξo)= f(gn · xo) dn and ϕ (g · xo)= ϕ(gk · ξo) dk . ZN ZK As mentionedb before, in the K bi-iinvariant setting this type of Radon transform had already appeared (with an extra factor aρ) in the work of Harish-Chandra [HC58] via the map f 7→ Ff . It also appeared in the fundamental work by Gelfand and Graev [GG59, GG62] where they introduced the “horospherical method”. In this section we introduce the Radon transform on X and discuss some of its properties. It should be noted that Helgason introduced the Radon transform in [H63a, H63b] but the Fourier transform only appeared later in [H65a], see also [H66b]. We have seen that the Fourier transform on X gives a unitary isomorphism ⊕ 2 dλ L (X) ≃ (πλ, Hλ) 2 + |c(λ)| Za whereas the Fourier transform in the A-variable gives a unitary isomorphism ⊕ 2 L (Ξ) ≃ (πλ, Hλ) dλ . Zia As the representations πλ and πwλ, w ∈ W , are equivalent this has the equivalent formulation L2(Ξ) ≃ (#W )L2(X) . In hindsight we could construct an intertwining operator from the following sequence of maps dλ L2(X) → L2 K/M × ia∗, → L2(K/M × ia∗, dλ) → L2(Ξ) |c(λ)|2 obtained with b = k · bo from the sequence: 1 1 f 7→ f (b) 7→ f(λ, b) 7→ F −1 f(·,b) (a) =: Λ(f)(ka · ξ ) . λ c(λ) A c(·) o This ideae plays a role ine the inversion of thee Radon transform, but instead we start with the Fourier transform on X given by (2.7). Then using b = k · bo we have

f(λ, b) = f(x)eλ,b(x) dx ZX −1 λ−ρ e = f(g · xo)a(g l) dg ZX −1 λ−ρ = f(lg · xo)a(g ) , dg ZX −λ+ρ = f(lan · xo)a dnda A N Z Z ρ = FA((·) R(f)(l(·))(λ) . ONTHELIFEANDWORKOFS.HELGASON 13

Here R(f)= fˆ is the Radon Transform from before. Thus we obtain that the factorization of the unitary G map discussed above, namely the Fourier transform on L2(X) is followed by the Radon transform, which is then followed by the Abelian Fourier transform on A, all this modulo the application of the pseudo-diﬀerential operator J corresponding to the Fourier multiplier 1/c(λ). Following [H65a] and [H70, p. 41 and p. 42] we therefore deﬁne the operator Λ by

−ρ ρ Λ(f)(ka · ξo)= a Ja(a f(ka · ξo)) . We then get [H65a, Thm. 2.1] and [H70]: Theorem . ∞ 2.7 Let f ∈ Cc (X). Then

#W |f(x)|2 dx = |ΛR(f)(ξ)|2 dξ ZX ZΞ 1 2 ∞ and f 7→ #W ΛR(f) extends to an isometry into L (X). Moreover, for f ∈ Cc (X) 1 f(x)= (ΛΛ∗fˆ)∨(x) . #W With inversion in hand, in [H63b] and [H73b] Helgason obtains the key properties of the Radon transform needed for the analysis of invariant diﬀerential operators on X. First we have the compatibility with a type of Harish-Chandra isomorphism: Theorem 2.8. There exists a homorphism Γ: D(X) → D(Ξ) such that for f ∈ Cc(X) we have R(Df)=Γ(D)R(f). Then using the Paley-Wiener Theorem for the symmetric space X Helgason generalizes his earlier support theorem. Theorem . ∞ 2.9 ( [H73b]) Let f ∈ Cc (X) satisfy the following conditions: (1) There is a closed ball V in X. (2) The Radon transform f(ξ)=0 whenever the horocycle ξ is disjoint from V . Then f(x)=0 for x∈ / V . b He now has all the pieces of the proof of his surjectivity result. Theorem 2.10. [H73b, Thm. 8.2] Let D ∈ D(X). Then DC∞(X)= C∞(X). The support theorem has now been extended to noncompact reductive symmetric spaces by Kuit [K11].

2.6. The Poisson Transform. On a symmetric space X the use of the Pois- son transform has a long and rich history. But into this story ﬁts a very precise and important contribution - the “Helgason Conjecture". In this section we recall brieﬂy the background from Helgason’s work leading to this major result. 2 ∞ Let g ∈ L (K/M) and f ∈ Cc (X). Recall from Theorem 2.5 that the Fourier 2 ∗ dλ 2 W transform can be viewed as having values in L (ia , #W |c(λ)|2 ,L (K/M)) . Denote ∗ the Fourier transform on X by FX (f)(λ) = fλ and by FX its adjoint. Then we

e 14 G.ÓLAFSSONANDR.J.STANTON

∗ evaluate FX as follows ∗ hFX (f),gi = hf, FX (g)i dλ = f(x) e−λ,b(x)g(b) db dx . ∗ |c(λ)|2 ZX Zia ZK/M ! The function inside the parenthesis is the Poisson transform

Pλ(g)(x) := e−λ,b(x)g(b) db. (2.11) ZK/M Helgason had made the basic observation that the functions eλ,b are eigenfunctions for D(X), i.e., there exists a character χλ : D(X) → C such that

Deλ,b = χλ(D)eλ,b . Indeed, they are fundamental to the construction of the Helgason Fourier transform. Here they form the kernel of the construction of eigenfunctions. Let ∞ Eλ(X) := {f ∈ C (X) | (∀D ∈ D(X)) Df = χλ(D)f} . (2.12)

Since D ∈ D(X) is invariant the group G acts on Eλ. This defines a continuous ∞ representation of G where Eλ carries the topology inherited from C (X). We have ∞ ∞ Pλg ∈Eλ and Pλ : Hλ = C (K/M) →Eλ is an intertwining operator. In the basic paper [H59] we have seen that various properties of joint solutions of operators in D(X) are obtained. In hindsight, one might speculate about eigenvalues different than 0 for operators in D(X), and what properties the eigenspaces might have. In fact, such a question is first formulated precisely in [H70] where several results are obtained. Are the eigenspaces irreducible? Do the eigenspaces have boundary values? What is the image of the Poisson transform on various function spaces? In [H70] Helgason observed that, as b 7→ e−λ,b(x) is analytic, the Poisson transform extends to the dual A′(K/M) of the space A(K/M) of analytic functions on K/M. Recall the Harish-Chandra c-function c(λ) and denote by ΓX (λ) the denominator of c(λ)c(−λ). The Gindikin- Karpelevich formula for the c-function gives an ∗ explicit formula for ΓX (λ) as a product of Γ-functions. An element λ ∈ aC is simple ∞ if the Poisson transform Pλ : C (K/M, ) →Eλ(X) is injective. Theorem 2.11 (Thm. 6.1 [H76]). λ is simple if and only if the denominator of the Harish-Chandra c-function is non-singular at λ. This result was used by Helgason for the following criterion for irreducibility: Theorem 2.12 (Thm 9.1, Thm. 12.1, [H76]). The following are equivalent:

(1) The representation of G on Eλ(X) is irreducible. (2) The principal series representation πλ is irreducible. −1 (3) ΓX (λ) 6=0. In [H76] p.217 he explains in detail the relationship of this result to [K75]. With irreducibility under control, Helgason turns to the range question. In [H76] for all symmetric spaces of the non-compact type, generalizing [H70, Thm. 3.2] for rank one spaces, he proves ONTHELIFEANDWORKOFS.HELGASON 15

Theorem 2.13. Every K-finite function in Eλ(X) is of the form Pλ(F ) for some K-finite function on K/M. In [H70, Ch. IV,Thm. 1.8] he examines the critical case of the Poincaré disk. Utilizing classical function theory on the circle he shows that eigenfunctions have boundary values in the space of analytic functionals. This, coupled with the aforementioned analytic properties of the Poisson kernel allow him to prove ′ ∗ Theorem 2.14. Eλ(X)= Pλ(A (T)) for λ ∈ ia Those results initiated intense research related to finding a suitable compactifi- cation of X compatible with eigenfunctions of D(X); to hyperfunctions as a suitable class of objects on the boundary to be boundary values of eigenfunctions; to the generalization of the Frobenius regular singular point theory to encompass the operators in D(X); and finally to the analysis needed to treat the Poisson transform and eigenfunctions on X. The result culminated in the impressive proof by Kashi- wara, Kowata, Minemura, Okamoto, Oshima and Tanaka [KKMOOT78] that the Poisson transform is a surjective map from the space of hyperfunctions on K/M onto Eλ(X), referred to as the “Helgason Conjecture". 2.7. Conical Distributions. Let X be the upper halfplane C+ = {z ∈ C | Re (z) > 0} = SL(2, R)/SO(2). A horocycle in C is a circle in X meeting the real line tangentially or, if the point of tangency is ∞, real lines parallel to the x-axis. It is easy to see that the horocycles are the orbits of conjugates of the group 1 x N = x ∈ R . 0 1

This leads to the deﬁnition for arbitrary symmetric spaces of the noncompact type:

Definition 2.15. A horocycle in X is an orbit of a conjugate of N. Denote by Ξ the set of horocycles. Using the Iwasawa decomposition it is easy to see that the horocycles are the subsets of X of the form gN · xo. Thus G acts transitively on Ξ and Ξ= G/MN. As we saw before ⊕ 2 2 L (Ξ) ≃ (πλ, Hλ) dλ ≃ (#W )L (X) (2.13) ∗ Zia the isomorphism being given by

ρ −λ ρ φλ(g) := [a ϕ(ga · ξo)]a da = FA([(·) ℓg−1 ϕ]|A)(λ) . ZA The descriptionb of L2(Ξ) ≃ (#W )L2(X) suggests the question of relating K invariant vectors with MN invariant vectors. But, as MN is noncompact, it follows from the theorem of Howe and Moore [HM79] that the unitary representations Hλ, λ ∈ ia∗ do not have any nontrivial MN-invariant vectors. But they have MN-ﬁxed distribution vectors as we will explain. ∞ Let (π, Vπ) be a representation of G in the Fréchet space Vπ. Denote by Vπ ∞ the space of smooth vectors with the usual Fréchet topology. The space Vπ is invariant under G and we denote the corresponding representation of G by π∞. The ∞ −∞ −∞ ∞ C conjugate linear dual of Vπ is denoted by Vπ . The dual pairing Vπ ×Vπ → , −∞ is denoted h·, ·i. The group G acts on Vπ by hπ−∞(a)Φ, φi := hΦ, π∞(a−1)φi . 16 G.ÓLAFSSONANDR.J.STANTON

The reason to use the conjugate dual is so that for unitary representations (π, Vπ) we have canonical G-equivariant inclusions ∞ −∞ Vπ ⊂ Vπ ⊂ Vπ . For the principal series representations we have more generally by (2.6) G-equivariant −∞ embeddings Hλ¯ ⊂ H−λ . −∞ MN Assume that there exists a nontrivial distribution vector Φ ∈ (Vπ ) . Then ∞ ∞ −∞ we define TΦ : Vφ → C (Ξ) by TΦ(v; g · ξo) = hπ (g)Φ, vi. Similarly, if T : ∞ ∞ Vπ → C (Ξ) is a continuous intertwining operator we can define a MN-invariant ∞ C distribution vector ΦT : Vπ → by hΦT , vi = T (v; ξo). Clearly those two maps are inverse to each other. The decomposition of L2(Ξ) in (2.13) therefore suggests −∞ MN that for generic λ we should have dim(Hλ ) = #W . As second motivation for studying MN-invariant distribution vectors is the following. Let (π, Vπ) be an irreducible unitary representation of G (or more generally −∞ MN ∞ an irreducible admissible representation) and let Φ, Ψ ∈ (Vπ ) . If f ∈ Cc (Ξ) −∞ ∞ −∞ then π (f)Φ is well defined and an element in Vπ . Hence hΨ, π (f)Φi is a well defined MN-invariant distribution on Ξ and all the invariant differential differential operators on Ξ coming from the center of the universal enveloping algebra act on this distribution by scalars. A final motivation for Helgason to study MN-invariant distribution vectors is the construction of intertwining operators between the representations (πλ, Hλ) and (πwλ, Hwλ), w ∈ W . This is done in Section 6 in [H70] but we will not discuss this here but refer to [H70] as well as [S68, KS71, KS80, VW90] for more information. We now recall Helgason’s construction for the principal series represenations 2 (πλ, Hλ). For that it is needed that Hλ = L (K/M) is independent of λ and ∞ ∞ ∗ ∗ Hλ = C (K/M). Let m ∈ NK(a) be such that m M ∈ W is the longest element. Then the Bruhat big cell, Nm¯ ∗AMN, is open and dense. Define aλ−ρ if g = n m∗aman ∈ Nm∗MAN ψ (g)= 1 2 (2.14) λ 0 if otherwise. −∞ If Re λ> 0 then ψλ ∈ H−λ¯ is an MN-invariant distribution vector. Helgason −∞ then shows in Theorem 2.7 that λ 7→ ψλ ∈ H−λ¯ extends to a meromorphic family of ∗ distribution vectors on all of aC. Similar construction works for the other N-orbits NwMAN, w ∈ W , leading to distribution vectors ψw,λ. Denote by D(Ξ) the algebra of G-invariant differential operators on Ξ. Then H 7→ DH extends to an isomorphisms of algebras S(a) ≃ D(Ξ), see [H70, Thm. 2.2]. Definition 2.16. A distribution Ψ (conjugate linear) on G is conical if (1) Ψ is MN-biinvariant. (2) Ψ is an eigendistribution of D(Ξ).

The distribution vectors ψw,λ then leads to conical distributions Ψw,λ and it is shown in [H70, H76] that those distributions generate the space of conical distributions for generic λ. ∗ ∞ ′ For λ ∈ aC let Cc (Ξ)λ (with the relative strong topology) denote the joint λ−ρ distribution eigenspaces of D(Ξ) containing the function g · ξo 7→ a(x) . Then G ∞ ′ acts on Cc (Ξ)λ and according to [H70, Ch. III, Prop. 5.2] we have: ONTHELIFEANDWORKOFS.HELGASON 17

Theorem . ∞ ′ 2.17 The representation on Cc (Ξ)λ is irreducible if and only if πλ is irreducible. 2.8. The Wave Equation. Of the many invariant differential equations on X the wave equation frequently was the focus Helgason’s attention. We shall discuss some of this work, but will omit his later work on the multitemporal wave equation [H98a, HS99]. 2 n ∂ Rn Rn Let ∆ = j=1 2 denote the Laplace operator on . The wave-equation ∂xj on Rn is the CauchyP problem ∂2 ∂ ∆Rn u(x, t)= u(x, t) u(x, 0) = f(x), u(x, 0) = g(x) (2.15) ∂t2 ∂t ∞ where the initial values f and g can be from Cc (X) or another “natural” function ∞ Rn space. Assume that f,g ∈ Cc ( ) with support contained in a closed ball BR(0) of radius R> 0 and centered at zero. The solution has a finite propagation speed in the sense that u(x, t)=0 if kxk− R ≥ |t|. The Huygens’ principle asserts that u(x, t)=0 for |t|≥kxk + R. It always holds for n > 1 and odd but fails in even ∞ R dimensions. It holds for n =1 if g ∈ Cc ( ) with mean zero. This equation can be considered for any Riemannian or pseudo-Riemannian manifold. In particular it is natural to consider the wave equation for Riemannian symmetric spaces of the compact or noncompact type. Helgason was interested in the wave equation and the Huygens’ principle from early on in his mathematical career, see [H64, H77, H84a, H86, H92a, H98]. One can probably trace that interest to his friendship with L. Ásgeirsson, an Icelandic mathematican who studied with Courant in Göttingen and had worked on the Huygens’ principle on Rn. One can assume that in (2.15) we have f = 0 and for simplicity assume that g is K-invariant. Then u can also be taken K-invariant. It is also more natural to consider the shifted wave equation ∂2 ∂ (∆ + kρk2)u(x, t)= u(x, t) u(x, 0)=0, u(x, 0) = g(x) (2.16) X ∂t2 ∂t There are three main approaches to the problem. The first is to use the Helgason Fourier transform to reduce (2.16) to the differential equation d2 d u(iλ,t)= −kλk2u(λ, t) , u(λ, 0)=0 and u(λ, 0) = g(λ) (2.17) dt2 dt for λ ∈ ia∗. From the inversion formula we then get e e e e e 1 sin kλkt dλ u(x, t)= g(λ)ϕλ(x) . #W ∗ kλk |c(λ)|2 Zia One can then use the Paley-Wiener Theoreme to shift the path of integration. Doing that one might hit the singularity of the c(λ) function. If all the root multiplicities are even, then 1/c(λ)c(−λ) is a W -invariant polynomial and hence corresponds to an invariant differential operator on X. Another possibility is to use the Radon transform and its compatibility with invariant operators 2 R((∆ + kρk )f)|A = ∆AR(f)|A then use the Helgason Fourier transform, and finally the Euclidean result on the Huygens’ principle. This was the method used in [ÓS92]. 18 G.ÓLAFSSONANDR.J.STANTON

Finally, in [H92a] Helgason showed that sin kλkt = e (x) dτ (x)= ϕ (x) dτ (x) kλk iλ,b t −λ t ZX ZX for certain distribution τt and then proving a support theorem for τt. The result is [ÓS92, H92a]: Theorem 2.18. Assume that all multiplicities are even. Then Huygens’s principle holds if rankX is odd. It was later shown in [BÓS95] that in general the solution has a speciﬁc exponential decay. In [BÓ97] it was shown, using symmetric space duality, that the Huygens’ principle holds locally for a compact symmetric spaces if and only it holds for the noncompact dual. The compact symmetric spaces were then treated more directly in [BÓP05]. Acknowledgements. The authors want to acknowledge the work that the referee did for this paper. His thorough and conscientious report was of great value to us for the useful corrections he made and the helpful suggestions he oﬀered.

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Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803 USA E-mail address: [email protected]

Department of Mathematics, Ohio State University, Columbus, OH, 43210 USA E-mail address: [email protected]